{
  "id": "1610.02593",
  "version": "v1",
  "published": "2016-10-08T22:39:15.000Z",
  "updated": "2016-10-08T22:39:15.000Z",
  "title": "Superlinearity of geodesic length in 2$D$ critical first-passage percolation",
  "authors": [
    "Michael Damron",
    "Pengfei Tang"
  ],
  "comment": "23 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "First-passage percolation is the study of the metric space $(\\mathbb{Z}^d,T)$, where $T$ is a random metric defined as the weighted graph metric using random edge-weights $(t_e)_{e\\in \\mathcal{E}^d}$ assigned to the nearest-neighbor edges $\\mathcal{E}^d$ of the $d$-dimensional cubic lattice. We study the so-called critical case in two dimensions, in which $\\mathbb{P}(t_e=0)=p_c$, where $p_c$ is the threshold for two-dimensional bond percolation. In contrast to the standard case $(<p_c)$, the distance $T(0,x)$ in the critical case grows sub linearly in $x$ and geodesics are expected to have Euclidean length which is superlinear. We show a strong version of this super linearity, namely that there is $s>1$ such that with probability at least $1-e^{-\\|x\\|_1^c}$, the minimal length geodesic from $0$ to $x$ has at least $\\|x\\|_1^s$ number of edges. Our proofs combine recent ideas to bound $T$ for general critical distributions, and modifications of techniques of Aizenman-Burchard to estimate the Hausdorff dimension of random curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-08T22:39:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical first-passage percolation",
      "geodesic length",
      "superlinearity",
      "critical case grows sub",
      "dimensional cubic lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}